Problem: $ C = \left[\begin{array}{rr}4 & 2 \\ 0 & -2 \\ 5 & 0\end{array}\right]$ $ w = \left[\begin{array}{r}4 \\ -1\end{array}\right]$ What is $ C w$ ?
Solution: Because $ C$ has dimensions $(3\times2)$ and $ w$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(3\times1)$ $ C w = \left[\begin{array}{rr}{4} & {2} \\ {0} & {-2} \\ \color{gray}{5} & \color{gray}{0}\end{array}\right] \left[\begin{array}{r}{4} \\ {-1}\end{array}\right] = \left[\begin{array}{r}? \\ ? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ w$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ w$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ w$ , and so on. Add the products together. $ \left[\begin{array}{r}{4}\cdot{4}+{2}\cdot{-1} \\ ? \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ w$ and add the products together. $ \left[\begin{array}{r}{4}\cdot{4}+{2}\cdot{-1} \\ {0}\cdot{4}+{-2}\cdot{-1} \\ ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{4}\cdot{4}+{2}\cdot{-1} \\ {0}\cdot{4}+{-2}\cdot{-1} \\ \color{gray}{5}\cdot{4}+\color{gray}{0}\cdot{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}14 \\ 2 \\ 20\end{array}\right] $